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Helmholtz, Duffing and Helmholtz-Duffing Oscillators: Exact Steady-State Solutions

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Abstract

This work presents an analytic technique aimed at designing the external excitation of linear and nonlinear oscillators so that a prescribed form of their steady-state response can be achieved. The technique exploits the exact analytic solutions of the oscillator response having quadratic and/or cubic nonlinearities. Both single-frequency and multi-frequency responses are considered. Examples of possible applications are provided in terms of virtual experiments.

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References

  1. Kovacic, I., Brennan, M.J. (eds.): The Duffing Equations: Nonlinear Oscillators and their Behaviour. Wiley, Chichester (2011)

    MATH  Google Scholar 

  2. Duffing, G.: Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung, Heft 41/42. Vieweg, Braunschweig (1918) (in German)

    Google Scholar 

  3. Nayfeh, A.H., Mook, D.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  4. Hsu, C.S.: On the application of elliptic functions in nonlinear forced oscillations. Q. Appl. Math. 17, 393–407 (1960)

    Article  Google Scholar 

  5. Rakaric, Z., Kovacic, I., Cartmell, M.P.: On the design of external excitations in order to make nonlinear oscillators respond as free oscillators of the same or different type. Int. J. Nonlinear Mech. 94C, 323–333 (2017)

    Article  Google Scholar 

  6. Kovacic, I.: On the response of purely nonlinear oscillators: an Ateb-type solution for motion and an Ateb-type external excitation. Int. J. Nonlinear Mech. 92, 15–24 (2017)

    Article  Google Scholar 

  7. Kovacic, I., Zukovic, M.: Coupled purely nonlinear oscillators: normal modes and exact solutions for free and forced responses. Nonlinear Dyn. 87, 713–726 (2017)

    Article  Google Scholar 

  8. Kovacic, I.: Externally excited undamped and damped linear and nonlinear oscillators: exact solutions and tuning to a desired exact form of the response. Int. J. Nonlinear Mech. 102, 72–81 (2018)

    Article  Google Scholar 

  9. Kovacic, I., Gatti, G.: Some benefits of using exact solutions of forced nonlinear oscillators: theoretical and experimental investigations. J. Sound Vib. 436, 310–326 (2018)

    Article  Google Scholar 

  10. Rand, R.H.: Using computer algebra to handle elliptic functions in the method of averaging. In: Noor, A.K., Elishakoff, I., Hulbert, G. (eds.) Symbolic Computations and Their Impact on Mechanics, vol. 205, pp. 311–326. American Society of Mechanical Engineers, PVP (1990)

    Google Scholar 

  11. Byrd, P., Friedman, M.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1954)

    Book  Google Scholar 

  12. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover Publications, New York (1965)

    MATH  Google Scholar 

  13. Gatti, G., Brennan, M.J.: Inner detached frequency response curves: an experimental study. J. Sound Vib. 396, 246–254 (2017)

    Article  Google Scholar 

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Acknowledgements

The first author acknowledges support of the Ministry of Education and Science of Serbia, grant ON174028.

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Correspondence to Gianluca Gatti .

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Kovacic, I., Gatti, G. (2020). Helmholtz, Duffing and Helmholtz-Duffing Oscillators: Exact Steady-State Solutions. In: Kovacic, I., Lenci, S. (eds) IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems. ENOLIDES 2018. IUTAM Bookseries, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-030-23692-2_15

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  • DOI: https://doi.org/10.1007/978-3-030-23692-2_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-23691-5

  • Online ISBN: 978-3-030-23692-2

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